PDE-aware deep learning for inverse problems in cardiac electrophysiology

25/2021

PDE-aware deep learning for inverse problems in cardiac electrophysiology

Monday 26th April 2021

Tenderini, R.; Pagani, S.; Quarteroni, A.; Deparis S.

In this work, we present a PDE-aware deep learning (DL) model, named Space-Time Reduced Basis Deep Neural Network (ST-RB-DNN), for the numerical solution to the Inverse Problem of Electrocardiography. The main feature of the proposed neural network (NN) is that it both leverages data availability and exploits the knowledge of a physically- based mathematical model, expressed by means of partial differential equations (PDEs), to carry out the task at hand. The goal is to estimate the epicardial potential field from measurements of the electric potential at a discrete set of points on the body surface. Such a problem has become central in biomedical research, providing the theoretical basis for Electrocardiographic Imaging (ECGI), but it is extremely hard to solve because of its ill-posedness. The employment of deep learning techniques in this context is made difficult by the low amount of clinical data at disposal (small data regime), as measuring cardiac potentials requires invasive procedures. Suitably exploiting the underlying physically- based mathematical model allowed to circumvent the data availability issue and led to the development of fast-training and low-complexity PDE-aware DL models. In particular, physical-awareness has been pursued by means of two elements: the projection of the epicardial potential onto a Space-Time Reduced subspace, spanned by the numerical solutions of the governing PDEs, and the inclusion of a tensorial Reduced Basis (RB) solver of the Forward Problem in the network architecture. Numerical tests have been conducted only on synthetic data, obtained via a Full Order Model (FOM) approximation of the problem at hand, and two variants of the model have been addressed. Both proved to be accurate, up to an average l1-norm relative error on epicardial activation maps of ? 3.5%, and both could be trained in ? 10 min. Nevertheless, some improvements, mostly concerning data generation, are necessary in order to bridge the gap with clinical applications.